RECALL FIRST CLASS:

Q = K __Head Difference______Area Distance between Heads

Q = 0.01 cm 0.19 m 6cm 0.75cm 1 liter 86400sec = 1.17 liter ~ 1 liter sec 0.63 m 1000cm3 day day day

constant head

0.4 m 0.21 m

FINE SAND K=0.01cm/s 6 cm 0.63 m 0.75 cm

For Water to Move a driving force is needed

but it doesn't necessarily flow downhill nor from high Pressure to low Pressure

Consider a siphon What drives the flow in that case?

pressure is fluid elevation is greater higher

This car is uphill

1 In a isothermal system of uniform electrochemical composition:

Flow Proceeds from High to Low Hydraulic Head i.e. from locations of high to low mechanical energy

Total mechanical energy depends on Fluid Pressure, Gravity, and Motion 1 E = P + ρgz + ρv2 total 2

Divide by density to get energy per unit mass P v2 E = + gz + unit mass ρ 2

The Bernoulli Equation from Fluid Mechanics

States that for steady, laminar, flow of frictionless, incompressible fluid energy per unit mass [L/T]2 is a constant (in practice these assumptions can be relaxed) P v2 E = + gz + = constant unit mass ρ 2

2 Steady Flow? Not changing with time

Laminar Flow? Reynolds number reflects flow regime R < 100 laminar R > 1000 turbulent Between 100 to 1000 transitional Velocity *diameter particle Vd dynamic viscosity μ R = = υ = = kinematic viscosity υ fluid density ρ Viscosity - resistance of a fluid to flow More on this later

Frictionless Flow? Real fluids are viscous (they require energy to overcome friction loss)

Incompressible Fluid? Real fluids are compressible (their density changes with pressure)

3 Bernoulli’s Equation is useful for comparing components of mechanical energy ⎛ L ⎞ 2 M Weight units ⎜ 2 ⎟ P v ⎜ T ⎟ + gz + = constant ⎜ L2 ⎟ ρ 2 ⎜ ⎟ L L2 ⎝ ⎠ + L + Divide each term b y g to use units of M T 2 T 2 3 energy per unit weight : dimensions of length L P v2 ⎛ L ⎞ ⎜ M 2 ⎟ + z + = hydraulic head ⎜ T ⎟ 2 2 gρ 2g units of ⎜ L ⎟ L L ⎜ ⎟ 2 L 2 ⎝ ⎠ + T + T length L M L L T 2 L3 T 2 T 2 Ground water velocity is generally so low that the kinetic term can be ignored P + z = hydraulic head gρ

P hydraulic head = + z ρg P P = ρgh h = p p ρg hydraulic head = h + z p a pithtipe that hydraulic head = pressure head + z is open at all can be expressed in length the ends

When If the density comparing were 10% Hydraulic heads if the greater at one h h headhathead, h, at density of well, would h p p location water in increase or column h x,y,z is p decrease to z differs, reflect an h = h + z normalize to equivalent 0’ p the same hydraulic datum (often sea level) density head?

4 Head exists at every location in a body of water If no forces are exerted on it and we allow it to “come to rest” head will be the same everywhere

We call this condition hydrostatic

h p h p hp hp

z z z z 0’ datum

If we force water through the system Head will vary with space

dry

h p hp hp z

z zz 0’ datum

5 Bernoulli’s equation for total mechanical energy per unit mass (here, omitting the velocity term) defines the Force Potential P force potential = Φ = + gz ρ

and P = ρgh p ρgh so Φ = p + gz = gh + gz = g(h + z) ρ p p

and ttltotal hdhydrauli c hdhead h = h p + z so Φ = gh thus Φ and h are related by a constant

Connecting lines of equal head (potential)

Yields head map / equipotential surface

10ft

0ft Plan view – Head map

7ft 8ft 9ft

6 It may be multi-dimensional

Let’s use a field scale system DISCHARGE AREA head RECHARGE AREA head increases with depth decreases with depth 10,390’

10,000’ 8,000’ 6,000’ 4,000’ 2,000’ 0’ 20,000 ft ~4 miles 10,000 ft thickness horizontal gradient along the red flow line i = 390ft/20000ft ~0.02 i = 390ft/24000ft ~0.016

7 Flow in Porous Media static conditions

steady flow in a 2” diameter sand-filled tube

8 What are the elevation and pressure heads?

Total head 25 inches

Total head 17. 5 inches Pressure head Total head 21.2 inches 10.8 inches Pressure head 15.5 inches

Elevation 3.8 inches Pressure head 15.8 inches dtdatum Elevation 2 inches Elevation -5 inches

Recall Darcy’s Law velocity of flow through the sand column is: * directly proportional to the head difference at the ends of the column and * inversely proportional to the length of the column

⎛ h2 − h1 ⎞ VDarcy = −K⎜ ⎟ = −Ki ⎝ l ⎠ constant of proportionality hydraulic gradient, i (hydraulic conductivity if water) in direction of flow

OTHER NAMES: Darcy Velocity or Specific discharge LT-1 (i.e. discharge per unit area, although sometimes multiplied by thickness to express as discharge per unit width as L2T-1 )

9 Calculate K using data from the Apparatus What measurements will you need? What equation will you solve?

How will flow change if we rotate the sand column? What if we bend the column?

10 Recall: Darcy velocity is a DISCHARGE per unit AREA Q V = Darcy A Average Linear Velocity “as the crow flies” velocity through the pores this governs rate of pollutant movement

Q VD VD VInterstitial = = = Aφe φe effect ive porosity

Calculate Effective Porosity using data from Darcy Apparatus

What measurement s will you need?

What equation will you solve?

Dye moves through the tube It moves ~30 cm in 7.5 min

11 Aquifer permits appreciable amounts of groundwater to pass under normal field conditions (passes economic quantities of water)

Aquitard Low hydraulic conductivity does not pass siggynificant amounts of water but may store water

K ---- HYDRAULIC CONDUCTIVITY when the fluid is water

The range of values spans many orders of magnitude: Gravel ~1x102cm/sec Unfractured Crystalline Rock ~ 1x10-11cm/sec

k --- PERMEABILITY the capacity of a porous medium to transmit fluid

12 MEASUREMENT OF K

FIELD TESTS - AQUIFER TESTS

LABORATORY - PERMEAMETERS problems not representative large rock mass disturbed samples orientation of sample

often knowing K to an order of magnitude is satisfactory and may be all that is obtainable within temporal and financial constraints

Hydraulic Conductivity K velocity units L/T or LT-1 FIELD COEFFICIENT OF HYDRAULIC CONDUCTIVITY describes K in terms of the rate of flow of water in gallons per day through a cross secti onal area of 1 square f oot under a hydraulic gradient of 1 at a temp of 60 degrees F

1ft GAL 1ft day 1ft 2 1GAL 1ft ft day 1ft3 ft cm ~ 0.13 ~ 3.5x10−5 ft 2 7.48GAL day sec

13 TRANSMISSIVITY T = K m, T units L2/T L2T Theis, 1935 FIELD COEFFICIENT OF TRANSMISSIVITY rate of flow of water in gal per day through a vertical strip of aquifer 1 ft wide, exten ding th e f ull sat urat ed h ei ght of th e aquif er under a hydraulic gradient of 100% or 1 ft per ft at prevailing field temperature

1ft GAL 1ft day b(ft) ft

1GAL 1ft day 1ft3 ft 2 cm2 ~ 0.13 ~ 1x10−7 ft 7.48GAL day sec

Very important T = Kb

Transmissivity = Hydraulic Conductivity * Aquifer Thickness

14 K hydraulic conductivity applies to a material only for water passing through it, and at that, watftilt&ter of particular temp & pressure ( iii.e. viscosit it)y) units of LT-1 e.g. cm sec-1

k intrinsic ppyermeability a characteristic of the medium independent of the fluid units of L2 e.g. cm2

k intrinsic permeability L M Kμ k = : T LT : L2 ρg M L L3 T 2 M μ = dynamic viscosity : LT M ρ = density : L3 L g = acceleration of gravity : T 2

15 k intrinsic permeability may be expressed in darcies 1 darcy is the permeability that will lead to a specific discharge, q, of 1cm/sec for a fluid with a viscosity of 1 centipoise under a hydraulic gradient that makes the term: ⎛ ρg dh ⎞ 1atm ⎜ ⎟ = ⎝ dl ⎠ cm

k ⎛ ρg dh ⎞ q = ⎜ ⎟ μ ⎝ dl ⎠

Let’s convert darcies to fundamental units of length squared

k ⎛ ρg dh ⎞ q = ⎜ ⎟ Recall Viscosity? μ ⎝ dl ⎠ Viscosity - resistance of a fluid to flow

(i.e. resistance to: pouring, deforming under shear stress, layers moving past each other) dynamic viscosity is the tangential force per unit area required to move one horizontal plane with respect to the other at unit velocity when maintained a unit distance apart by the fluid)

dynamic viscosity units: N s m-2 or Pa s or kgm-1s-1 note:1Pa-s=1N s m-2=1kg m s-1 or: gcm-1s-1 or dyne s cm-2 or poise note: 1poise = 1gcm-1s-1 = 1dyne s cm-2 = 0.1Pa-s

dynamic viscosity of water at 68.4oF (20.2oC) is 1centipoise dynamic viscosity μ kinematic viscosity =υ = = fluid density ρ

kinematic viscosity units: m2s-1 or Stokes 1St=10-4m2s-1 Since the specific gravity of water at 68.4oF (20.2oC) is almost 1g cm-3, kinematic viscosity of water at 68.4oF is for all practical purposes 1centiStoke (cSt)

16 k ⎛ ρg dh ⎞ q = ⎜ ⎟ k intrinsic permeability μ ⎝ dl ⎠ To convert darcies to cm2: 1cm 1darcy ⎛1atm ⎞ by definition : = ⎜ ⎟ sec 1centipoise ⎝ cm ⎠

1cm 1centipoise cm =1darcy sec 1atm

0.01g 1.01325x10−6 g 1centipoise = 1atm = cm sec cmsec2

0.01g cm 1cm cm sec cm g cm cm sec2 darcy = = 9.869x10−9 =1x10−8 cm2 sec 1.01325x10−6 g cm sec2 g cm sec2

To convert 1 darcy to K cm/sec for water:

1g 980cm 9.869x10−9 cm2 kρg 3 2 K = = cm sec μ 1.01x10−2 g cm sec

g cm cm sec cm = 9.6x10−4 cm2 =1x10−3 cm3 sec2 g sec

17 DARCY'S LAW - summary of basics details and assumptions dh V = −K or Q = KiA D dl NOT VALID FOR:

*very high velocities where turbulent conditions might prevail

*extremely low velocities in fine grained materials where there may be a threshold for which no flow occurs

*compressible fluids (recall we divded total energy by a constant density to define hydraulic head as energy per unit mass for Bernoulli’s equation)

*a discontinuum or variable properties (see next slide REV)

* pressure & elevation are not the only driving forces (see subsequent slide on other driving forces)

Darcy's is a macroscopic law we assign uniform constant K to the entire porous medium mass

REPRESENTATIVE ELEMENTARY VOLUME REV a macro-continuum, below this volume there is no single value of a given parameter that can represent the material K

REV

Sample size

18 *other phenomena contribute to the energy gradient that drives flow

temperature electric currents chemical variations

hence we should really write: dh dT dC dV V = −L − L − L − L 1 dl 2 dl 3 dl 4 dl

temperature & chemical gradients make water flow but flowing water will transport heat & dissolved chemicals this causes the temperature & chemical gradients that drive water flow to change which causes the rate of water flow to change which changes the rate of heat and chemical transport by the water etc this is coupled flow

SUMMARIZING ASSUMPTIONS OF DARCY'S LAW

*LAMINAR FLOW

*ABOVE THRESHOLD V, IF IT EXISTS

*THE FLUID IS INCOMPRESSIBLE

*CONTINUUM & K IS CONSTANT (in space and time)

*PRESSURE/ELEVATION HEAD IS ONLY DRIVING FORCE

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